Find the mistake (AM GM ineqality)

Question

According to AM GM inequality

$$\frac{\left(x^3-\frac{1}{x}-\frac{1}{x}+\frac{2}{x}\right)}{4}\ge\left[(x^3)\left(-\frac{1}{x}\right)\left(-\frac{1}{x}\right)\left(\frac{2}{x}\right)\right]^\frac{1}{4}$$

$$\Rightarrow x^3 \ge 2^\frac{9}{4}\;\;\;\;;\; \forall\; x\in \mathbb R^+ $$ which is not true

Answer 1

Your use of the AM-GM is equivalent to saying something like $$\frac{2 + (-1) + (-1)}{3} \ge \sqrt[3]{2(-1)(-1)}.$$ It doesn't work because we require each term to be a nonnegative real number.

Answer 2

The $GM-AM$ said the following

Let $$a,b>0 \, \, \text{Then the following inequality holds} \sqrt{ab}< \frac{a+b}{2}$$

as @Benjamin Wang remark your RHS is non positive.